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Zero CR-curvature equations for Levi degenerate hypersurfaces via Pocchiola’s invariants
Alexander Isaev1
1 Mathematical Sciences Institute, Australian National University, Canberra, Acton, ACT 2601, Australia
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 28 (2019) no. 5, pp. 957-976.

Dans nos articles précédents, nous avons étudié les hypersurfaces tubes dans l’espace euclidien complexe de dimension trois, qui sont 2-non-dégénérées et uniformément Levi dégénérées de rang 1. Nous avons montré en particulier que l’annulation de la courbure CR d’une telle hypersurface est équivalente à une équation de Monge par rapport à une des variables. Dans cet article nous dérivons cette équation par une approche alternative plus rapide, en utilisant deux invariants découverts par S. Pocchiola. Nous étudions également les invariants de Pocchiola dans le cas rigide et donnons une classification partielle des hypersurfaces rigides 2-non-dégénérées uniformément Levi dégénérées de rang 1 à courbure CR nulle.

In articles [8, 9] we studied tube hypersurfaces in ℂ 3 that are 2-nondegenerate and uniformly Levi degenerate of rank 1. In particular, we showed that the vanishing of the CR-curvature of such a hypersurface is equivalent to the Monge equation with respect to one of the variables. In the present paper we provide an alternative shorter derivation of this equation by utilizing two invariants discovered by S. Pocchiola. We also investigate Pocchiola’s invariants in the rigid case and give a partial classification of rigid 2-nondegenerate uniformly Levi degenerate of rank 1 hypersurfaces with vanishing CR-curvature.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/afst.1618
Classification : 32V05, 32V20, 32W20, 35J96, 34A05, 34A26
Mots clés : CR-curvature, tube and rigid hypersurfaces, the Monge–Ampère equation, the Monge equation, Pocchiola’s invariants
Alexander Isaev 1

1 Mathematical Sciences Institute, Australian National University, Canberra, Acton, ACT 2601, Australia
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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Alexander Isaev. Zero CR-curvature equations for Levi degenerate hypersurfaces via Pocchiola’s invariants. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 28 (2019) no. 5, pp. 957-976. doi : 10.5802/afst.1618. https://afst.centre-mersenne.org/articles/10.5802/afst.1618/

[1] Mohamed Salah Baouendi; Linda Preiss Rothschild; François Trèves CR structures with group action and extendability of CR functions, Invent. Math., Volume 82 (1985) no. 2, pp. 359-396 | DOI | Numdam | MR | Zbl

[2] Elie Cartan Sur la géométrie pseudo-conforme des hypersurfaces de l’espace de deux variables complexes: I, Ann. Mat. Pura Appl., Volume 11 (1933) no. 1, pp. 17-90 II, Ann. Scuola Norm. Sup. Pisa. 1 (1932), p. 333–354. | DOI | MR | Zbl

[3] Vladimir Ezhov; Gerd Schmalz Explicit description of spherical rigid hypersurfaces in ℂ 2 , Complex Anal. Synerg., Volume 1 (2015), 2, 10 pages | DOI | Zbl

[4] Gregor Fels; Wilhelm Kaup CR-manifolds of dimension 5: a Lie algebra approach, J. Reine Angew. Math., Volume 604 (2007), pp. 47-71 | MR | Zbl

[5] Wei G. Foo; Joël Merker Differential {e}-structures for equivalences of 2-nondegenerate Levi rank 1 hypersurfaces M 5 ⊂ℂ 3 (2019) (https://arxiv.org/abs/1901.02028)

[6] Hervé Gaussier; Joël Merker A new example of a uniformly Levi degenerate hypersurface in ℂ 3 , Ark. Mat., Volume 41 (2003) no. 1, pp. 85-94 erratum Ark. Mat. 45 (2007), p. 269–271

[7] Alexander V. Isaev Spherical tube hypersurfaces, Lecture Notes in Mathematics, 2020, Springer, 2011, xii+220 pages | MR | Zbl

[8] Alexander V. Isaev Affine rigidity of Levi degenerate tube hypersurfaces, J. Differ. Geom., Volume 104 (2016) no. 1, pp. 111-141 | DOI | MR | Zbl

[9] Alexander V. Isaev On the CR-curvature of Levi degenerate tube hypersurfaces, Methods Appl. Anal., Volume 23 (2016) no. 4, pp. 317-328 | MR | Zbl

[10] Alexander V. Isaev; Dmitri Zaitsev Reduction of five-dimensional uniformly Levi degenerate CR structures to absolute parallelisms, J. Geom. Anal., Volume 23 (2013) no. 3, pp. 1571-1605 | DOI | MR | Zbl

[11] Daniela Kraus; Oliver Roth Conformal metrics, Topics in modern function theory (Ramanujan Mathematical Society Lecture Notes Series), Volume 19, Ramanujan Mathematical Society, 2013, pp. 41-83 | MR | Zbl

[12] Joseph M. Landsberg Differential-geometric characterizations of complete intersections, J. Differ. Geom., Volume 44 (1996) no. 1, pp. 32-73 | DOI | MR | Zbl

[13] J. W. Lasley On Monge’s Differential Equation, Am. Math. Mon., Volume 43 (1936) no. 5, pp. 284-286 | DOI | MR | Zbl

[14] Costantino Medori; Andrea Spiro The equivalence problem for five-dimensional Levi degenerate CR manifolds, Int. Math. Res. Not., Volume 2014 (2014) no. 20, pp. 5602-5647 | DOI | MR | Zbl

[15] Joël Merker; Samuel Pocchiola Explicit absolute parallelism for 2-nondegenerate real hypersurfaces M 5 ⊂ℂ 3 of constant levi rank 1, J. Geom. Anal. (2018) | DOI

[16] Samuel Pocchiola Explicit absolute parallelism for 2-nondegenerate real hypersurfaces M 5 ⊂ℂ 3 of constant Levi rank 1 (2013) (https://arxiv.org/abs/1312.6400)

[17] Curtis Porter The local equivalence problem for 7-dimensional, 2-nondegenerate CR manifolds whose cubic form is of conformal unitary type (2015) (https://arxiv.org/abs/1511.04019)

[18] Curtis Porter; Igor Zelenko Absolute parallelism for 2-nondegenerate CR structures via bigraded Tanaka prolongation (2017) (https://arxiv.org/abs/1704.03999)

[19] Nancy Stanton A normal form for rigid hypersurfaces in ℂ 2 , Am. J. Math., Volume 113 (1991) no. 5, pp. 877-910 | Zbl

[20] Vitaly Ushakov The explicit general solution of trivial Monge–Ampère equation, Comment. Math. Helv., Volume 75 (2000) no. 1, pp. 125-133 | DOI | MR | Zbl

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